Difference between revisions of "Energy in a signal"

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(Energy of a Signal)
(Energy of a Signal)
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By [[Rayliegh's Theroem]],
 
By [[Rayliegh's Theroem]],
 
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
 
: <math> <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
This implies that the energy of a signal can be found by integrating the square of the fourier transform of the signal,
+
This implies that the energy of a signal can be found by the fourier transform of the signal,
 
: <math> W = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
 
: <math> W = \int_{-\infty}^{\infty} |V(f)|^2\,df </math>
 
This page is far from complete please feel free to pick up where it has been left off.
 
This page is far from complete please feel free to pick up where it has been left off.

Revision as of 20:20, 10 October 2006

Definition of Energy

Energy is the ability or potential for something to create change. Scientifically energy is defined as total work done by a force. Work can be mathematically calculated as the line integral of force per infinatesimal unit distance,

 W = \int \mathbf{F} \cdot \mathrm{d}\mathbf{s}

Power represents a change in energy.

 P(t) = \frac{dW}{dt}

This means we can also write energy as

 W = \int_{-\infty}^{\infty} P(t)\,dt

Energy of a Signal

From circuit analysis we know that the power generated by a voltage source is,

P(t) = {v^2(t) \over R}

Assuming that R is 1 then the total energy is just,

W = \int_{-\infty}^\infty |v|^2(t) \, dt

This can be written using bra-ket notation as

 <v(t) | v(t)> \! or  <v|v> \!

By Rayliegh's Theroem,

 <v|v> = \int_{-\infty}^{\infty} |V(f)|^2\,df

This implies that the energy of a signal can be found by the fourier transform of the signal,

 W = \int_{-\infty}^{\infty} |V(f)|^2\,df

This page is far from complete please feel free to pick up where it has been left off.